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The way I see it, 10-12 days should give you plenty of time to see pretty much whatever you want, as long as you are always heading towards your destination. Basically, as long as you don't zig-zag too much, you should be fine.

Unfortuantly, however, that might make it difficult to fit both Denver and Rushmore or Glacier into this trip.

I'd say you should go one of two ways. After visiting your friend in Chicago, either stay on I-90 and see the Badlands, Rushmore/Black Hills, Yellowstone, and Glacier, before making tracks back Southwest. Or continue on to Denver, and spend some time exploring the mountains and parks in Colorado and Utah. Either option is a good one, but I think you'll have to make a choice in which one sounds most appealing to you.